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Comments on some recently proposed experiments that

should distinguish Bohmian mechanics from quantum

mechanics

W. Struyve, W. De Baere

Laboratory for Theoretical PhysicsUnit for Subatomic and Radiation PhysicsProeftuinstraat 86, B–9000 Ghent, Belgium

E–mail: willy.debaere@rug.ac.be, ward@odiel.rug.ac.be

Abstract

Recently Ghose [1–3] and Golshani and Akhavan [4–6] claimed to havefound experiments that should be able to distinguish between StandardQuantum Mechanics and Bohmian Mechanics. It is our aim to show thatthe claims made by Ghose, Golshani and Akhavan are unfounded.

1 Introduction

According to Standard Quantum Mechanics (SQM), the complete descriptionof a physical system is provided by its wave function. In Bohmian Mechanics(BM)1 the standard description of quantum phenomena, by means of the wave-function ψ, is enlarged by considering particles that follow definite tracks inspace-time (dependent on the initial conditions). These positions of a particleon these tracks act as the hidden variables of SQM. The positions of the parti-cles are hidden because BM is constructed in a way to give the same statisticalpredictions as SQM if a measurement is performed. This is accomplished byassuming the probability distribution for an ensemble in BM to be the same asthe quantum mechanical distribution. This distribution is called the quantum

equilibrium distribution (see section 2).Yet, recently Ghose [1–3] and Golshani and Akhavan [4–6] (short GGA)

proposed some experiments that should be able to distinguish between SQMand BM at the level of individual detections. It is the aim of the present work,however, to show that the claims made by GGA are unfounded. Moreover itshould be clear that, with the quantum equilibrium hypothesis in mind, onecannot obtain a distinguishment between SQM and BM. Only a modified orextended Bohmian theory can yield experimentally observable differences withSQM. This stresses once more the fact that BM is nothing more than a possible(causal) interpretation of SQM, as is clearly stated already in the beginning byBohm [8, 9].

1For a mathematical review see [7].

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2 Quantum equilibrium hypothesis

In SQM a physical system is described in configuration space by its wavefunctionψ(x1, . . . ,xn, t), dependent on n 3-vectors xj . This wavefunction obeys theSchrodinger equation

ih∂ψ(x1, . . . ,xn, t)

∂t= Hψ(x1, . . . ,xn, t) (1)

Given an initial wavefunction ψ(x1, . . . ,xn, 0) this equation can be solved togive a unique solution ψ(x1, . . . ,xn, t). When a position measurement is per-formed on an ensemble of identically prepared systems (all described by thesame wavefunction), the probability P (Q1, . . . ,Qn, t0) of making a joint detec-tion at a certain time t0 of the n-particles at positions Q1, . . . ,Qn in physicalspace is given by

P (Q1, . . . ,Qn, t0) = ψ∗(Q1, . . . ,Qn, t0)ψ(Q1, . . . ,Qn, t0) (2)

In BM, SQM is considered as an incomplete theory. Apart from a wave-function (obeying (1)) one introduces additional (hidden) variables to describethe physical system. These hidden variables are n vectors that have to beinterpreted as actual position vectors Xk(t) associated with n particles in 3-dimensional physical space. According to BM these vectors are also the positionvectors revealed in a position measurement. This is contrary to SQM where noparticles exist as localized entities, i.e. as entities that have position vectors,until a position measurement is performed.

Bohm [8, 9] obtained the laws of motion for the particles by giving a new in-terpretation to the real and imaginary part of the Schrodinger equation. The realpart is interpreted as a classical Hamilton-Jacobi equation with an additionalquantum mechanical potential, the quantum potential. This interpretation leadsto the following differential equations for the position vectors Xk(t)

dXk

dt=

h

mkIm

ψ∗(x1, . . . ,xn, t)∇kψ(x1, . . . ,xn, t)

|ψ(x1, . . . ,xn, t)|2

∣

∣

∣

xj=Xj

(3)

where mk is the mass of the kth particle. Once we have a solution for equation(1), equation (3) can be solved given the initial positions Xk(0). In this way then actual position vectors Xk(t) of the particles are uniquely determined. If wethen consider an ensemble of systems, all described by the same wavefunction,then this ensemble determines a probability distribution ρ(X1, . . . ,Xn, t) of theactual position vectors of the n particles. This is the distribution that would beobtained, according to BM, when a position measurement on an ensemble wereperformed. If we want BM to give the same predictions as SQM in a positionmeasurement, then the probability distribution P of SQM in equation (2), hasto be the same as the probability distribution ρ of BM, i.e. we must have

ρ(X1, . . . ,Xn, t) = |ψ(X1, . . . ,Xn, t)|2 (4)

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for all times t. If this equality is assumed, and this is what is done in BM [8, 9],the imaginary part of the Schrodinger equation

∂|ψ|2

∂t+

∑

k

∇k · (vk|ψ|2) = 0 (5)

with

vk =h

mkIm

ψ∗∇kψ

|ψ|2(6)

is the continuity equation, describing the conservation of the probability densityof the particles. In fact it is sufficient that we assume

ρ(X1, . . . ,Xn, t0) = |ψ(X1, . . . ,Xn, t0)|2 (7)

at a certain time t0 (for example at t = 0) because both |ψ|2 and ρ satisfythe continuity equation. Thus, as far as predictions involving particle positionsare concerned, BM is in complete accordance with SQM if the initial particlepositions Xk(0) are distributed according to |ψ(X1, . . . ,Xn, 0)|2 in the ensem-ble. This is what is called the quantum equilibrium hypothesis (QEH) by Durr,Goldstein and Zanghı[10] and the distibution is called the quantum equilibrium

distribution. Because every measurement is in fact a position measurement, it isclear that there can never be an experimental difference between BM and SQM,as a result of the QEH. This implies also that, despite of definite trajectories inBM, we can only predict and verify relative frequencies. So an individual eventcannot be studied independently from the ensemble.

The only difference that remains between BM and SQM is an interpreta-tional one. In BM ρ(Q1, . . . ,Qn, t0) is interpreted as the probability of theparticles really being at the positions Q1, . . . ,Qn at time t0 whereas in SQMP (Q1, . . . ,Qn, t0) is the probability of the particles being detected at the posi-tions Q1, . . . ,Qn at time t0.

The reason why we cannot distinguish the two theories is that we cannotobserve a particle without disturbing its movement. I.e. we cannot obtain knowl-edge of the position of the particle without changing its wavefunction (this is thecollapse in SQM). This changing wavefunction leads then to changing particlevelocities (as follows from (3)), leaving a disturbed system. The best exampleof this is the diffraction at a slit: the smaller the slit (i.e. the better we try toget the initial position in the slit), the wider the scattering angle. In this way aquantum mechanical measurement is very different from a measurement in clas-sical mechanics, where trajectories of objects are generally accepted becauseone can infer successive positions of an object without disturbing its motion, forexample by using light that scatters from the object.

We want to remark that the QEH was already postulated by Bohm in orderto assure complete equivalence between BM and SQM. So BM does not provideus with new experimentally verifiable predictions, but instead gives us a broaderconceptual framework that may serve as a basis for new or modified mathema-tical formulations for the description of physical systems. In such theories theQEH will evidently break down and this is what Bohm meant in [9]:

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“An experimental choice between these two interpretations cannotbe made in a domain in which the present mathematical formulationof the quantum theory is a good approximation, but such a choice isconceivable in domains, such as those associated with dimensions ofthe order of 10−13 cm, where the extrapolation of the present theoryseems to break down and where our suggested new interpretation canlead to completely different kinds of predictions.”

Such modifications and extensions of BM are for example given by Bohm himselfin [8, 11–13].

3 Outline and discussion of the experiments

The proposed experiments of Ghose [1, 2] and Golshani and Akhavan [4, 5] makeuse of a pair of identical, non-relativistic, bosonic particles labeled 1 and 2. Theparticles emerge pair by pair (so there is only one pair in the device at a time)from a source placed in front of a screen with two identical slits A and B, withcoordinates (0,±Y ). The particles are simultaneously diffracted by the twoslits. We can suppose the wavefunction of the system, in the x− y plane, afterdiffraction, to be of the form

ψ(x1, y1, x2, y2, t) = N [ψA(x1, y1, t)ψB(x2, y2, t) + ψB(x1, y1, t)ψA(x2, y2, t)](8)

where ψA is the diffracted wave coming from slit A, and ψB is the one comingfrom slit B. This wavefunction can be considered to describe particles going todifferent slits. The case in which a pair of particles can “travel” through oneslit at a time ([4, 5]) is not considered here. The two particles are then simul-taneously detected at a fixed screen parallel with the y-axis. If we suppose thedetectors to be idealized pointdetectors, then SQM gives the following proba-bility for detecting the pair of particles (in the ensemble) at time t0 at positionsy1 = Q1 and y2 = Q2 on the screen:

P (Q1, Q2, t0) = |ψ(y1, y2, t)|2∣

∣

y1=Q1,y2=Q2,t=t0= |ψ(Q1, Q2, t0)|

2 (9)

If the detectors detect over regions ∆Q1 and ∆Q2 then the probability of jointdetection is given by

P12 =

∫ Q1+∆Q1

Q1

∫ Q2+∆Q2

Q2

dy1dy2|ψ(y1, y2, t)|2 (10)

According to the QEH these probabilities must be the same for BM.We will now first look at Ghose’s explanation of how a distinguishment can

be achieved [1, 2]. Using plane waves for ψA and ψB [2] or more general usingthe natural symmetry of the setup (the slits are symmetrically placed aroundthe x-axis) together with the bosonic symmetry [1], Ghose finds that the y-

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coordinates of the Bohmian trajectories (calculated with (3)) satisfy2

y1 + y2 = 0 or y1(t) + y2(t) = y1(0) + y2(0) (11)

If the two particles depart initially symmetrically about the x-axis, i.e.

y1(0) + y2(0) = 0 (12)

then the particles will remain symmetrically about the x-axis for all times. Thisleads Ghose to state the following3

“. . . consider then the Bohm ensemble to be built up of single pairsof particle trajectories arriving at the screen at different instants oftime ti such that the joint probability of detection is given by

P ∗12 = lim

N→∞

N∑

i=1

1

Nδ(0)

∫

dy1dy2P (y1, y2, ti)

×δ(y1 − y1(ti))δ(y2 − y2(ti))δ(y1(ti) + y2(ti))

= limN→∞

N∑

i=1

1

NP (y1(ti),−y1(ti), ti) = 1 (13)

where the constraint (12) has been taken into account. Every termin the sum represents only one pair of trajectories arriving at thescreen at the points (y1(ti),−y1(ti)) at time ti . . . ”

Ghose then concludes that when the detectors are placed symmetrically aboutthe x-axis, they will record coincidence counts, just as SQM predicts. But if theyare placed asymmetrically, the joint detection of every pair, and hence also theirtime average, will produce a null result which is not predicted by SQM. InsteadSQM gives the probability P12 (in equation (10)) for joint detection. Thiswould be the incompatibility between SQM and BM. However, it is clear thatP (y1(ti),−y1(ti), ti) 6= 1 for every chosen couple (y1(ti),−y1(ti)), because of theidentity P ≡ |ψ|2. In addition, it is impossible to fix the intitial positions ofevery pair in order to have y1(0)+y2(0) = 0, without making measurements. Wewill discuss this in depth by using Gaussian waves. Following thereby Golshaniand Akhavan [4, 5], the wavefunction in the x-direction is a plane wave and inthe y-direction the slits generate Gaussian profiles

ψA,B(x, y, t) = (2πσ2t )−1/4e−(±y−Y −uyt)2/4σ0σt+i[ky(±y−Y −uyt/2)]

×ei[kxx−Et/h] (14)

where

σt = σ0(1 +iht

2mσ20

) (15)

2Similar equations are obtained with spherical waves for ψA and ψB [14, 15] or Gaussianwaves (see below).

3Taken from [2] adjusted with the factor 1/N and adapted to our notations.

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ux,y =hkx,y

m(16)

E =1

2mu2

x (17)

The motion in the x-direction is irrelevant and will be suppressed. If one definesy = (y1 + y2)/2 as the centre of mass in the y-direction, then one finds by using(3), (8) and (14) that4

y =(h/2mσ2

0)2

1 + (h/2mσ20)

2t2yt (18)

which yields after integration

y(t) = y(0)√

1 + (h/2mσ20)

2t2 (19)

If at t = 0 the centre of mass of the particles is exactly on the x-axis (i.e.y(0) = 0) then the centre of mass will remain on the x-axis for all time. Soaccording to BM the particles will always be detected symmetrically about thex-axis if y(0) = 0 for each pair of particles. Then Golshani and Akhavan arguein the line of Ghose that this property leads to a distinguishment when the timeensemble (13) is considered.

There are several remarks in order:

1. Although the probability of detecting every pair of particles symmetri-cally about the x-axis is equal to one if equation (12) is assumed valid,P (y1(ti),−y1(ti), ti) cannot be equal to 1 for every pair (y1(ti),−y1(ti))at time ti, because P (y1, y2, t) = |ψ(y1, y2, t)|

2 (together with (8)). So P ∗12

cannot be equal to 1 .

2. We can also not restrict the particles to depart initially symmetricallyabout the x-axis without changing the wavefunction, because |ψ|2 doesnot contain the constraint y(0) = 0, or put in another way ψ does notrestrict y(0) from being different from zero. The initial positions of theparticles (in BM) are distributed according to the absolute square of thewavefunction. So the initial coordinate of the centre of mass y(0) is in thesame way distributed. Given a system with wavefunction ψ then we cannotdetermine the initial conditions better than provided by the wavefunctionψ (i.e. we may not violate the QEH). Thus if we want y(0) to be zero thenwe have to change the wavefunction ψ by making measurements.

The fact that the initial positions have to be distributed according to theabsolute square of the wavefunction has also been noted by Marchildon[16], who goes on by demonstrating in specific cases that BM and SQMindeed lead to the same predictions if no restriction on y(0) is supposed.

3. As an example we will show what happens if we take the slits very narrowin order to make sure that the particles depart symmetrically. Small slits

4Taken from [4, 5].

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imply a small σ0 in (14). Looking at (19) one sees that y(t) becomes large,although y(0) is small, if time increases. This leads to a high probabilityof assymmetrical detection on the screen. In order to assure that y(t) ≈y(0) one could propose to make m large. So the particles would remainsymmetrically in BM. But then ψ in equation (8) contains sharply peakedgaussians that do not spread rapidly, because then uy ≪ and σt ≪. Soalso SQM then predicts a symmetrical, joint detection.

4. In addition to the described experiment, Golshani and Akhavan [4, 5] pro-posed another one which uses selective detection. Suppose that y(0) = δand that each detector is placed on one side of the x-axis. If one detectoris placed approximately on the x-axis then, from equation (19), the otherdetector will never detect the other particle within a distance L ≥ δ. Ifevery pair in the ensemble is restricted in this way, there would be anobservable difference. But because Golshani and Akhavan again have tosuppose restricted initial positions, they again violate the QEH.

They even claim [5] that one can make ∆y(0) ≪ σ0, despite of the QEH.But little calculation shows that if the overlap of the wavefuncion is negli-gible at t = 0, i.e. σ0 is small compared to Y (so crossterms in expectationvalues can be dropped), then ∆y(0) = 1√

2σ0. One cannot adjust y0 inde-

pendently of σ0, again as a consequence of the QEH.

5. In [6] Golshani and Akhavan propose an experiment similar to the onedescribed above. Because the distinguishment between the two theories isbased on the same grounds, the same arguments apply to that experimentas well.

6. If it would be the purpose of GGA to contest the QEH, then they shouldhave to propose an alternative mathematical formulation of BM. But thisis not what is suggested by GGA in [1–6] and [17]. By the way, contestingthe QEH could be done with less complicated experiments.

4 Ergodicity

According to Ghose [3, 15] the incompatibity between BM and SQM in the two-slit experiment described above arises from the non-ergodic properties of theBohmian description of the system, whereas SQM would be ergodic for everysystem. This was the reply of Ghose to the paper of Marchildon [16]. We willnow show that Ghose’s definitions of ergodicity for SQM can be adopted to BMas well and on the other hand that his proof of the non-ergodicity of BM canbe refuted. For definitions and theorems concerning ergodicity we refer to thebook of Arnold and Avez.

Let us first look at the proof of Ghose [3] for the ergodicity of SQM of anarbitrary two particle system with wavefunction ψ(x1, x2, t). We can write ψ in

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the basis of orthonormal energy eigenfunctions φn(x1, x2)

ψ(x1, x2, t) =∑

n

cne−iEnt/hφn(x1, x2) (20)

With the time and space average for any observable F , denoted respectively F ∗

and F , Ghose proves that

F ∗ = limT→∞

1

T

∫ T

0

dt

∫

dx1dx2ψ∗(x1, x2, t)Fψ(x1, x2, t)

=∑

n

|cn|2

∫

dx1dx2φ∗n(x1, x2)F φn(x1, x2)

= F (21)

This defines F ∗ and F . The equality of space and time averages for an arbitraryobservable F induces ergodicity for SQM. Note that F is not the entire spaceaverage of F at t = 0, but only an average over the diagonal elements. So,regardless the question wether or not it is usefull to introduce ergodicity in thisway for quantum mechanics, the proof above can be adopted without problemsto the case of BM as well. This is done by noting that the complete spaceaverage of the observable F

∫

dx1dx2ψ∗(x1, x2, t)F ψ(x1, x2, t) (22)

in (21) equals the space average in BM of the “local expectation value” F (x1, x2, t)of the hermitian operator F defined by

F (x1, x2, t) = Reψ∗(x1, x2, t)Fψ(x1, x2, t)

ψ∗(x1, x2, t)ψ(x1, x2, t)(23)

Hence F can be interpreted as a property of the particle in BM (see [7]). Thespace average in BM is then

∫

dx1dx2P (x1, x2, t)F (x1, x2, t)

= Re

∫

dx1dx2ψ∗(x1, x2, t)Fψ(x1, x2, t)

=

∫

dx1dx2ψ∗(x1, x2, t)F ψ(x1, x2, t) (24)

where the hermiticity of F has been taken into account. So, the space average inBM equals the space average in SQM. This introduces ergodicity, as introducedby Ghose, for BM as well.

The non-ergodicity of BM was proved in [17], by showing that the 2-slitsystem is decomposable, thereby again assuming the condition (12). But re-linquishing this condition can lead to ergodicity of BM. Non-ergodicity of the

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Bohmian system would imply that the space average P12 (from (10)) and thetime average P ∗

12 (from (13)) are not equal. The explanation of Ghose is that

P ∗12 = 0 and P12 6= 0 (25)

if the detectors are placed sufficiently asymmetrical. Apart from the discussionon the validity of expression (13), already held in the previous section, we wantto quote a theorem of G.D. Birkhoff and A.J. Khinchin [18] which states that fora dynamical system (M,µ, φt) and f ∈ L1(M,µ) a complex valued µ-summablefunction on M , the following equality holds

∫

M

f∗dµ = f (26)

This is clearly in contradiction with (25).In [3] Ghose provides us with an additional example of a non-ergodic system

in BM. The system is the quantum mechanical analog of two coupled harmonicoscillators. Ghose comes to the same conclusions as for the 2-slit systems, usingthe same arguments. So my counter-arguments apply as well.

5 Conclusion

The cause of the different predictions made by BM and SQM, is that boththeories different wavefunctions were used for the calculations of the probabilitydistributions. BM used a wavefunction with restricted initial positions whileSQM did not. Thus a new wavefunction has to be introduced, if we wantto restrict the intitial conditions, which leads again to the same predictionsas SQM. The trajectories that are hidden in BM cannot be revealed withoutdisturbing the system.

Acknowledgements:

WS acknowledges financial support from the F.W.O. Belgium.

References

[1] P. Ghose, quant-ph/0001024.

[2] P. Ghose, quant-ph/0003037.

[3] P. Ghose, quant-ph/0103126.

[4] M. Golshani and O. Akhavan, quant-ph/0009040.

[5] M. Golshani and O. Akhavan, quant-ph/0103100.

[6] M. Golshani and O. Akhavan, quant-ph/0103101.

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[7] P.R. Holland, “The Quantum Theory of Motion”, Cambridge UniversityPress, Cambridge (1993).

[8] D. Bohm, Phys. Rev. 85 , 166, (1952).

[9] D. Bohm, Phys. Rev. 85 , 180, (1952).

[10] D. Durr, S. Goldstein and N. Zanghı, Jour. Stat. Phys. 67, 843, (1992).

[11] D. Bohm, Phys. Rev. 89 , 89, (1952).

[12] D. Bohm and J.P. Vigier, Phys. Rev. 96 , 208, (1954).

[13] D. Bohm and B.J. Hiley, “The Undivided Universe”, Routledge, New York,(1993), chapter 9.

[14] L. Marchildon, quant-ph/0007068.

[15] P. Ghose, quant-ph/0008007.

[16] L. Marchildon, quant-ph/0101132.

[17] P. Ghose, quant-ph/0102131.

[18] V.I. Arnold and A. Avez, “Ergodic Problems of Classical Mechanics”, W.A.Benjamin, Inc. (1968).

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